Contents I. Introduction
Fuzzy logic systems (FLSs) are good enough to perform any nonlinear dynamic system action due to its model free approach. A fuzzy adaptive control system that incorporates the expert information is a fuzzy logic system equipped with a training algorithm in which the fuzzy logic system is constructed from a collection of fuzzy IF-THEN rules, and the training algorithm adjusts the parameters of the fuzzy logic system to match the input/output data. [1]–[5]. Thus, adaptive control schemes can be extensive applications for a wide variety of industrial systems. In many applications of control engineering, processes to be controlled or simply monitored are located far from the computing unit and the measured data are transmitted through a low-rate communication system. In this situation, the task of reconstructing the system state should include output delay which is equivalent to that of the state prediction in the undelayed output case. The issue of state reconstructing of the present state in presence of time delays in system dynamic equation and/or in the measurement process has received increasing attention. This problem is not only interesting for the control, but also for the supervision and real-time monitoring of systems. However, most of the fuzzy control algorithms assumed nonlinear systems without delayed output for general design procedures to guarantee basic performance criteria. Thus, the problem of state reconstructing for nonlinear system with output delay is very important. State observation for systems with delay only in the state equation was proposed in [6], [7], for nonlinear systems with delays as well as output delay which are linearizable by additive output injection [8], and for nonlinear system state predictions with small output delay [9], [10]. In [11], the authors proposed a chain observer for known nonlinear dynamic systems with delayed output. Therefore, how to design an observer for an unknown nonlinear system with delayed output is a very important problem.
